Question

find the area of a regular hexagon with apothem length 3 cm. if necessary, write your answer in simplified radical form. area = cm²

Explanation:

Step1: Find the side length of the hexagon

For a regular hexagon, the relationship between the apothem \(a = 3\) cm and the side length \(s\) can be found using the formula for a 30 - 60 - 90 triangle (since a regular hexagon can be divided into six equilateral triangles, and the apothem is the height of one of the six equilateral triangles). If we consider the right - triangle formed by half of the side length \(x\), the apothem \(a\), and the radius \(r\) (where \(r = s\) for a regular hexagon). In a 30 - 60 - 90 triangle, \(\tan60^{\circ}=\frac{a}{x}\), and \(a = 3\) cm. Since \(\tan60^{\circ}=\sqrt{3}\), and \(x=\frac{s}{2}\), we have \(\sqrt{3}=\frac{3}{x}\), so \(x = \sqrt{3}\) cm. Then \(s = 2\sqrt{3}\) cm.

Step2: Calculate the perimeter \(P\) of the hexagon

The perimeter \(P\) of a regular hexagon with side length \(s\) is \(P=6s\). Substituting \(s = 2\sqrt{3}\) cm, we get \(P = 6\times2\sqrt{3}=12\sqrt{3}\) cm.

Step3: Use the formula for the area of a regular polygon

The formula for the area \(A\) of a regular polygon is \(A=\frac{1}{2}aP\), where \(a\) is the apothem and \(P\) is the perimeter. Substituting \(a = 3\) cm and \(P=12\sqrt{3}\) cm, we have \(A=\frac{1}{2}\times3\times12\sqrt{3}\).

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Answer:

\(18\sqrt{3}\)